Over the past two decades, numerous optimal scheduling algorithms for real-time systems on multiprocessor platforms have been proposed for the Liu & Layland task model. However, recent studies showed that even if optimal algorithms can theoretically schedule any feasible task set, suboptimal algorithms usually perform better when executed on real computation platforms. This can be explained by the runtime overheads that such optimal algorithms induce. We have observed that all current optimal online multiprocessor real-time scheduling algorithms are (completely or partially) based on the notion of fairness. The respect of this fairness can be the cause of numerous preemptions and migrations. We therefore propose a new algorithm -named U-EDF- which releases the property of fairness and instead use an EDFlike scheduling policy. The simulation results are really encouraging since they show that, in average, U-EDF produces less than one preemption and one migration per job released during the schedule. Furthermore, we strongly believe in the optimality of our algorithm since all tested task sets were correctly scheduled under U-EDF.